As we mentioned above, the coupling through the Stern-Gerlach force between the equation of motion Eq. (REF ) and spin dynamics Eq. (REF ) disappears if projected with \(\hat{k}^\mu \)  and \(\varepsilon ^\mu \) . In such a case the motion and spin precession is governed by only the TBMT equations of motion [1]}. The situation here is more complex than the neutral particle case presented in [2]} as the equations of motion gain additional terms proportional to particle charge. In the charged case, the projection of 4-velocity and laser polarization \(\varepsilon \cdot u(\tau )\)  is no longer a constant of motion. Instead we obtain after multiplying Eq. (REF ) by \(\varepsilon ^\mu \)
\(\varepsilon \cdot \dot{u} = \frac{d}{d\tau } (\varepsilon \cdot u) = \frac{e\mathcal {A}_0 \omega }{mc} k \cdot u(0) f^{\prime }(\xi )\;,\) 
